Optimal Signal Constellation Design for Ultra-High-Speed Optical Transport in the Presence of Phase Noise

ABSTRACT

A method to process applicable to coherent optical channels with either linear or nonlinear phase noise includes: splitting a received sequence of data into clusters of points according to a cumulative log-likelihood function from constellation obtained in a previous iteration; generating new constellation points by calculating a center of mass of the clusters of points; repeating until convergence or until a predetermined number of iterations has been reached to determine a signal constellation; and transmitting signals over the coherent optical channels with nonlinear phase noise using the disclosed signal constellation and LDPC-coded modulation concepts.

This application claims priority to Provisional Application 61/890,452filed Oct. 14, 2013, the content of which is incorporated by reference.

BACKGROUND

As a response to never ending high bandwidth demands, the IEEE hasratified its 40/100 Gb/s Ethernet Standard IEEE 802.3ba in June 2010.The deployment of 100 Gb/s Ethernet (GbE) has already started and it isexpected to accelerate in next few years. At these ultra-high datarates, the performance of fiber-optic communication systems is degradedsignificantly due to presence of various linear and nonlinearimpairments. To deal with those channel impairments novel advancedtechniques in modulation and detection, coding and signal processinghave been intensively studied. For carrier phase estimation (CPE), thealgorithmic DSP-based approaches are highly popular, and can becategorized into two broad categories data-aided and non-data-aided. TheMAP and ML approaches are particularly efficient in CPE; however, thecomplexity of such algorithms grows exponentially with the channelmemory. Even upon compensation of chromatic dispersion and nonlinearityphase compensation there will be some residual phase error. It has beenexperimentally verified that even in beyond 100 Gb/s transmission thedistribution of samples upon compensation of linear and nonlinearimpairments is still Gaussian-like with the residual phase error thatcan properly be modeled as a Markov process.

SUMMARY

In one aspect, a method to process coherent optical channels withnonlinear phase noise includes: splitting a received sequence of datainto clusters of points according to a cumulative log-likelihoodfunction from constellation obtained in a previous iteration; generatingnew constellation points by calculating a center of mass of the clustersof points; repeating until convergence or until a predetermined numberof iterations has been reached to determine a signal constellation; andtransmitting signals over the coherent optical channels with nonlinearphase noise using the signal constellation.

In another aspect, a signal constellation design method is disclosedwhich is applicable to coherent detection systems with residual phaseerror introduced by imperfect carrier phase estimator. Instead of usingthe Euclidian distance as optimization criterion that is optimum onlyfor additive white Gaussian noise (AWGN) channel, the system defines acumulative log-likelihood function and uses it as an optimizationcriterion instead. The optimization criterion is applicable to scenariosin which either linear or nonlinear phase noise dominates. The optimumsource distribution is obtained by maximizing the channel capacity,based on Arimoto-Blahut algorithm. The process can be considered as ageneralization of the optimum signal constellation design (OSCD) method.Since the method uses the cumulative log-likelihood function as theoptimization criterion, based on log-likelihood ratio (LLR) calculation,it has been named here LLR-based OSCD (LLR-OSCD) method. We alsodisclose an LDPC coded modulation scheme suitable for use in combinationwith constellations obtained by LLR-OSCD method, which is suitable foruse in situations when residual phase error is present in receiver. ThisLDPC coded modulation scheme represents the generalization of scheme.Monte Carlo simulations indicate that the LDPC-coded modulation schemesbased on signal constellations obtained by algorithm significantlyoutperform the corresponding LDPC-coded QAM.

Advantages of the preferred embodiments may include one or more of thefollowing. The optimum signal constellation design algorithm, namedLLR-OSCD, is suitable for receiver operating in the presence of residuallinear/nonlinear phase noise. The method employs the cumulativelog-likelihood function as an optimization criterion. The method isapplicable to both the channels with linear and nonlinear phase noise,and can easily be generalized to other channels. The LDPC-codedconstellations show better robustness to the residual phase noisecompared to the conventional QAM-based schemes. The code-modulationscheme, employing signal constellations from LLR-OSCD, is robust tolaser phase noise, imperfect CPE, and nonlinear phase noise introducedby fiber nonlinearities. The method is robust to cyclic slips. The MonteCarlo integration method to use to calculate the symbol likelihoods inthe presence of residual phase error. The method is applicable to both2-D and multidimensional signaling schemes. The optimized modulationschemes, when used in combination with LDPC coding, are more robust inthe presence of phase error than conventional LDPC-coded QAM. The methodcan also be used in both SMF and FMF applications.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A shows exemplary optimized 2D 16-ary signal constellations for a16-ary LLR-OSCD.

FIG. 1B show exemplary optimized 2D 16-ary signal constellations for a16-ary OSCD.

FIG. 2 shows an exemplary LDPC coded modulation scheme with Monte Carlointegration.

FIG. 3 show exemplary 2D 8-ary and 16-ary constellations for NL-OSCD.

FIG. 4 Uncoded BER vs. launch power for 8-ary NL-OSCD and 8-QAM.

FIG. 5 LDPC-coded BER vs. total transmission distance 8-ary NL-OSCD and8-QAM.

DESCRIPTION

A new signal constellation design algorithm applicable to coherentdetection systems with residual phase error introduced by imperfectcarrier phase estimator is disclosed. The cumulative log-likelihoodfunction is defined and used as an optimization criterion. Theoptimization criterion is applicable to scenarios in which either linearor nonlinear phase noise dominates. It can straightforwardly begeneralized to other scenarios. The optimum source distribution isobtained by maximizing the channel capacity, based on Arimoto-Blahutalgorithm. We also disclosed an LDPC coded modulation scheme suitablefor use in combination with constellations obtained by LLR-OSCDalgorithm, which is suitable for use in situations when residual phaseerror is present in receiver. The LDPC-coded LLR-OSCD modulation schemesshow much better robustness to phase noise compared to traditionalLDPC-coded QAM.

Optimal Signal Constellation Design for Linear Phase Noise Channel

In the presence of phase noise, instead of minimizing the mean squareerror, we define the cumulative log-likelihood function and get theoptimal signal constellation that maximizes this function. Namely, theEuclidean distance receiver is optimum only for the AWGN channel. Thestarting point in the algorithm is to use the conventionalArimoto-Blahut algorithm to determine the optimum source distributionfor the phase noise channel, and then generate the constellation samplesfrom this source. Then we run the LLR-OSCD method. After initialization(for example, conventional QAM constellation and CIPQ constellation canbe used for initialization), we split the received sequence intoclusters of points according to the cumulative log-likelihood functionfrom constellation obtained in previous iteration. New constellationpoints are generated by calculating the center of mass of such obtainedclusters. This procedure is repeated until convergence or until apredetermined number of iterations has been reached.

The LLR-OSCD method can be formulated as follows.

-   -   Initialization: Choose the signal constellation that will be        used for initialization. Let the size of constellation be M.    -   Generate the training sequence from the optimum source        distribution. Denote this sequence as {x_(j); j=0, . . . , n−1}.    -   Group the samples from this sequence into M clusters. The        membership to the cluster is determined based on LLR of sample        point and candidate signal constellation points from previous        iteration. Each sample point is assigned to the cluster with the        largest LLR. Given the m-th subset (cluster) with N candidate        constellation points, denoted as Â_(m)={y_(i); i=1, . . . , N},        find the likelihood function (LL) of partition P(Â_(m))={S_(i);        i=1, . . . , N}, as follows

$\begin{matrix}{{LL}_{m} = {{{LL}\left( \left\{ {{\hat{A}}_{m},{P\left( {\hat{A}}_{m} \right)}} \right\} \right)} = {n^{- 1}{\sum\limits_{k = 0}^{n - 1}\; {\max\limits_{y \in {\hat{A}}_{m}}\; {{LL}\left( {x_{k},y} \right)}}}}}} & (1)\end{matrix}$

The function LL(x_(j), y) is the cumulative log-likelihood functiondefined as

$\begin{matrix}{{{LL}\left( {x_{k},y} \right)} = {\frac{1}{NS}{\sum\limits_{i = 1}^{NS}\; {- \frac{\left\{ {x_{k\; 1} - {{Re}\left\lbrack {\left( {y_{1} + {y_{2}j}} \right)^{{- j} \times {PN}_{}}} \right\rbrack}} \right\}^{2} + \left\{ {x_{k\; 2} - {{Im}\left\lbrack {\left( {y_{1} + {y_{2}j}} \right)^{{- j} \times {PN}_{}}} \right\rbrack}} \right\}^{2}}{2\delta^{2}}}}}} & (2)\end{matrix}$

Where NS denotes the number of phase noise samples and the correspondingphase noise sample is denoted as PN_(i). x_(k1) and x_(k2) denote thefirst and second coordinates of the point x_(k). Similarly, y₁ and y₂denote the coordinates of the point y. The equation above is applicableto two-dimensional (2D) signal constellation design, but it canstraightforwardly be generalized for arbitrary dimensionality.

-   -   Determine the new signal constellation points as center of mass        for each cluster.    -   Repeat the steps 2)-3) until convergence.

This process is applicable to both linear and nonlinear phase noisechannels. As an illustration, in FIG. 1 we provide: (a) the 16-aryLLR-OSCD 2D constellation and (b) the 16-ary OSCD 2D constellation forcomparison. The results are obtained for phase noise and ASE noisedominated scenario, by setting the frequency offset×symbol durationproduct to 10⁻³.

Notice that the OSCD constellation is optimal for ASE noise dominatedchannel [3]. The optimal source for OSCD is Gaussian distribution andthe criterion for determining the membership for cluster is Euclideandistance squared. It is obvious that the result has three layers(central point is an individual layer) and has the circle shape.However, the constellation obtained by LLR-OSCD algorithm is optimizedfor the channel affected by phase noise and ASE noise simultaneously. Byhaving the pentagon-like shape at outer layer and moving oneconstellation point from inner layer to the pentagon layer, the 16-aryLLR-OSCD can deal better with the phase noise, and our simulation willconfirm this observation.

LDPC Coded Modulation Scheme Employing the LLR-OSCDs

The equivalent phase noise channel model, we mentioned above, is brieflydescribed below. The Wiener phase noise model can be used to describethe laser phase noise and imperfect carrier phase estimation (CPE),which is

θ_(k)=(θ_(k-1)+Δθ_(k))mod 2π  (3)

where θ_(k) denotes the residual phase error (at kth time instance) andΔθ_(k) denotes the zero-mean Gaussian process of varianceδ_(Δθ)=2πΔfT_(s), in which T_(s) is symbol duration and Δf denotes thelinewidth or frequency offset. We can also model the cyclic slips byMarkvo-like process of certain memory. The resulting noise process isGaussian-like distribution and the corresponding PDF is given by

$\begin{matrix}{{p\left( {{ra_{k}},\theta_{k}} \right)} = {^{- \frac{{{{r_{k} - {s{({a_{k},\theta_{k}})}}}}}^{2}}{N_{0}}}/\left( {\pi \; N_{0}} \right)}} & (4)\end{matrix}$

where s(a_(k), θ_(k))=e^(jθ) ^(k) [a_(k) ¹, a_(k) ², . . . , a_(k) ^(N)]and a_(k) ^(i) is the ith coordinate of transmitted symbol. We can alsouse histogram to estimate the conditional PDF when the channel is notGaussian. We will implement a new method called Monte Carlo integration,instead of numerical integration when estimating the log-likelihoodfunction, which is shown as

l(a)=log E _(θ){exp[l(a,θ)]}(5)

where l(a, θ) is the log-likelihood function for the transmitted symbolsand E_(θ) is the expectation average.

The LDPC coded modulation scheme we used in the simulation is shown inthe FIG. 2. The b independent data are first encoded by (n,k) LDPCencoder and written in row-wise fashion into b×n block interleaver. Thenthe LLR-OSCD mapper takes b bits to select a constellation point and 4Dmodulator will convert the coordinates to optical domain. At thereceiver side, the log-likelihood are be calculated by (5), after thecoherent detection and CPE. Once the bit LLRs are calculated, the LDPCdecoders perform decoding for b bits at the same time. The extrinsicinformation is then iterated between APP demapper and LDPC decoders.FIG. 2 shows the structure of the scheme for 2D signaling and thisscheme is also suitable for 4D transmission.

Nonlinear Phase Noise Model

Our channel model is nonlinear phase noise model with discreteamplification for the finite number of fiber spans. When the opticalsignal is periodically amplified by EDFAs, the nonlinear phase noise isunavoidably added to the optical signal and accumulated as the numberspans increases. For convenience, we consider the discrete memorylesschannel model, which is introduced in [4] and can be described asfollows:

Y=(X+Z)e ^(−jΦ) ^(NL)   (1)

where XεX is the channel input, Z is the total additive noise, and Y isthe channel observation. (In (1) j denotes the imaginary unit.) In eachfiber span, the overall nonlinear phase shift Φ_(NL) is given by

Φ_(NL)=∫₀ ^(L) γP(z)dz=γL _(eff) P  (2)

where P is the launch power and γ is the nonlinear Kerr-parameter. For afiber span length of L with attenuation coefficient of α, the powerevolution is described as P(z)=Pe^(−+z) effective length is defined as

$\begin{matrix}{L_{eff} = \frac{1 - ^{{- \alpha}\; L}}{\alpha}} & (3)\end{matrix}$

For a system with N_(A) fiber spans, the overall nonlinear phase noiseis given by:

$\begin{matrix}{\Phi_{NL} = {\gamma \; L_{eff}\left\{ {{{E_{0} + n_{1}}}^{2} + {{E_{0} + n_{1} + n_{2}}}^{2} + \cdots + {{E_{0} + n_{1} + \cdots + n_{N_{A}}}}^{2}} \right\}}} & (4)\end{matrix}$

where E₀ is the baseband representation of the transmitted electricfield, n_(k) is independent identically distributed zero-mean circularGaussian random complex variable with variance δ₀ ². The total additivenoise ate the end of all fiber segments has the variance δ²

E[Z²]=2N_(A)δ₀ ² and can be calculated as [5]

δ²=2n _(sp) hvαΔN _(A)  (5)

where all parameters are summarized in the Table I [6].

In this channel model, the variance of the phase noise is dependent onthe channel input and the channel is specified by the number of spans,transmission length, and the launch power.

Optimal Signal Constellation Design for the Nonlinear Phase NoiseChannel

In the presence of nonlinear phase noise, we can use an algorithmsimilar to OSCD algorithm [2] but now changing the optimizationcriterion from minimizing the mean square error to maximizing thecumulative log-likelihood function in order to get the optimal signalconstellation. The optimum source distribution for the nonlinear phasenoise channel can be obtained by Arimoto-Blahut algorithm and then wecan generate the training samples from this source. Then we run ourproposed algorithm, which is described below. In this algorithm weperform clustering of the constellation points generated by optimumsource based on cumulative log-likelihood function. New constellationpoints will be then obtained by calculating the center of mass of suchobtained clusters. This procedure is repeated until convergence or untila predetermined number of iterations has been reached.

The proposed algorithm, called here nonlinear optimum signalconstellation design (NL-OSCD), can be formulated as follows.

0) Initialization step. Choose the signal constellation that will beused for initialization and normalize the power of constellation totarget launch power P. (Both QAM and IPQ constellation can be used asinitialization.) Let the size of constellation be M.

1) Generate the training sequence from the optimum source distribution.Denote this sequence as {x_(j); j=0, . . . , n−1}.

2) The clustering step. Group the samples from this sequence into Mclusters. The membership to the cluster is determined based on thelog-likelihood ratio (LLR) of sample point and candidate signalconstellation points from previous iteration. Each sample point isassigned to the cluster with the largest LLR. Given the mth subset(cluster) with N candidate constellation points, denoted asÂ_(m)={y_(i); i=1, . . . , N}, find the log-likelihood (LL) function ofpartition P(Â_(m))={S_(i); i=1, . . . , N} as follows

$\begin{matrix}{{LL}_{m} = {{{LL}\left( \left\{ {{\hat{A}}_{m},{P\left( {\hat{A}}_{m} \right)}} \right\} \right)} = {n^{- 1}{\sum\limits_{k = 0}^{n - 1}\; {\max\limits_{y \in {\hat{A}}_{m}}\; {{LL}\left( {x_{k},y} \right)}}}}}} & (6)\end{matrix}$

The function LL(x_(j), y) is the cumulative log-likelihood functiondefined as

$\begin{matrix}{{{LL}\left( {x_{k},y} \right)} = {\frac{1}{NS}{\sum\limits_{i = 1}^{NS}\; {- \frac{\left\{ {x_{k\; 1} - {{Re}\left\lbrack {\left( {y_{1} + {y_{2}j}} \right)^{{- j} \times {PN}_{}}} \right\rbrack}} \right\}^{2} + \left\{ {x_{k\; 2} - {{Im}\left\lbrack {\left( {y_{1} + {y_{2}j}} \right)^{{- j} \times {PN}_{}}} \right\rbrack}} \right\}^{2}}{2\delta^{2}}}}}} & (7)\end{matrix}$

where NS denotes the number of phase noise samples and the correspondingnonlinear phase noise sample is denoted as PN_(i), which can begenerated using the channel model described in Section 2. x_(k1) andx_(k2) denote the first and second coordinates of the constellationpoint x_(k). Similarly, y₁ and y₂ denote the coordinates of the receivedvector (point) y. The equation above is applicable to two-dimensional(2D) signal constellation designs, but it can straightforwardly begeneralized for arbitrary dimensionality.

3) Determine the new signal constellation points as the center of massfor each cluster.

Repeat the steps 2)-3) until convergence.

As an illustration, in FIG. 3 we disclose the 2D signal constellationsfor 8-ary and 16-ary signaling obtained by NL-OSCD.

The signal constellation design method is applicable to coherentdetection systems with residual phase error introduced by imperfectcarrier phase estimator. Instead of using the Euclidian distance asoptimization criterion that is optimum only for additive white Gaussiannoise (AWGN) channel, the system defines a cumulative log-likelihoodfunction and use it as an optimization criterion instead. Theoptimization criterion is applicable to scenarios in which either linearor nonlinear phase noise dominates. The optimum source distribution isobtained by maximizing the channel capacity, based on Arimoto-Blahutalgorithm. The process can be considered as a generalization of theoptimum signal constellation design (OSCD) method. Since the method usesthe cumulative log-likelihood function as the optimization criterion,based on log-likelihood ratio (LLR) calculation, it has been named hereLLR-based OSCD (LLR-OSCD) method. We also an LDPC coded modulationscheme suitable for use in combination with constellations obtained byLLR-OSCD method, which is suitable for use in situations when residualphase error is present in receiver. This LDPC coded modulation schemerepresents the generalization of scheme. Monte Carlo simulationsindicate that the LDPC-coded modulation schemes based on signalconstellations obtained by algorithm significantly outperform thecorresponding LDPC-coded QAM.

Advantages may include one or more of the following. The optimum signalconstellation design algorithm, named LLR-OSCD, is suitable for receiveroperating in the presence of residual phase noise. The method employsthe cumulative log-likelihood function as an optimization criterion. Themethod is applicable to both the channels with linear and nonlinearphase noise, and can easily be generalized to other channels. TheLDPC-coded constellations show better robustness to the residual phasenoise compared to the conventional QAM-based schemes. Thecode-modulation scheme, employing signal constellations fromLLR-OSCD/NL-OSCD, is robust to laser phase noise, imperfect CPE, andnonlinear phase noise introduced by fiber nonlinearities. The method isrobust to cyclic slips. The Monte Carlo integration method to use tocalculate the symbol likelihoods in the presence of residual phaseerror. The method is applicable to both 2-D and multidimensionalsignaling schemes. The optimized modulation schemes, when used incombination with LDPC coding, are more robust in the presence of phaseerror than conventional LDPC-coded QAM. The method can also be used inboth SMF and FMF applications.

As an illustration, in FIGS. 4-5 we provide the results obtained byNL-OSCD method for 8-ary NL-OSCD, for optimized amplifier spacing,corresponding to aggregate data rate 2×25 Gs/s×3=150 Gb/s persingle-carrier. The FIG. 4 clearly indicates the existence of theoptimal launch power for total transmission distance of 3500 km for both8-ary NL-OSCD and 8-QAM. It can also be noticed that the nonlineartolerance of 8-NL-OSCD is better. NL-OSCD (−6, 2000) curve denotes thatthis constellation has been obtained by employing NL-OSCD with thelaunch power of −6 dBm and for total transmission distance of 2000 km.We can see that NL-OSCDs with different parameters have similar optimalpower around −1.5 dBm. The total transmission distance that can beachieved by employing NL-OSCDs and LDPC coding is around 9000 km, asshown in FIG. 5. After running the NL-OSCD algorithm for the launchpower of −1.5 dBm and total transmission distance of 9000 km, thetransmission distance can be extended to close to 11000 km. Notice thatthis result is obtained without NL compensation. This initial study willbe extended to larger constellation sizes.

This disclosure is intended to explain how to fashion and use variousembodiments in accordance with the technology rather than to limit thetrue, intended, and fair scope and spirit thereof. The foregoingdescription is not intended to be exhaustive or to be limited to theprecise forms disclosed. Modifications or variations are possible inlight of the above teachings. The embodiment(s) was chosen and describedto provide the best illustration of the principle of the describedtechnology and its practical application, and to enable one of ordinaryskill in the art to utilize the technology in various embodiments andwith various modifications as are suited to the particular usecontemplated. All such modifications and variations are within the scopeof the invention as determined by the appended claims, as may be amendedduring the pendency of this application for patent, and all equivalentsthereof, when interpreted in accordance with the breadth to which theyare fairly, legally and equitably entitled.

What is claimed is:
 1. A signal constellation design method to enabletransmission over coherent optical channels with nonlinear phase noise,comprising: splitting a received training sequence of data into clustersof points according to a cumulative log-likelihood function fromconstellation obtained in a previous iteration; generating newconstellation points by calculating a center of mass of the clusters ofpoints; repeating until convergence or until a predetermined number ofiterations has been reached to determine a signal constellation; andtransmitting signals over the coherent optical channels with nonlinearphase noise using the signal constellation such designed.
 2. The methodof claim 1, comprising estimating a log-likelihood function:l(a)=log E _(Φ) _(NL) {exp[l(a,Φ _(NL))]} where l(a, Φ_(NL)) is alog-likelihood function for transmitted symbols and E_(Φ) _(NL) is anexpectation average over a nonlinear phase Φ_(NL).
 3. The method ofclaim 1, comprising calculating log-likelihood after coherent detectionand carrier phase estimation (CPE).
 4. The method of claim 1, comprisingdecoding for b data LDPC coded streams at the same time and iteratingextrinsic information between an APP demapper and LDPC decoders.
 5. Themethod of claim 1, comprising encoding b independent data by an (n,k)LDPC encoder and writing in row-wise fashion into a b×n blockinterleaver.
 6. The method of claim 1, comprising performing Monte Carlointegration at a receiver.
 7. The method of claim 1, comprising optimumsource distribution used in the algorithm is generated by maximizing thechannel capacity based on Arimoto-Blahut method.
 8. The method of claim1, comprising applying a function LL(x_(j), y) as a cumulativelog-likelihood function defined as${{LL}\left( {x_{k},y} \right)} = {\frac{1}{NS}{\sum\limits_{i = 1}^{NS}\; {- \frac{\left\{ {x_{k\; 1} - {{Re}\left\lbrack {\left( {y_{1} + {y_{2}j}} \right)^{{- j} \times {PN}_{}}} \right\rbrack}} \right\}^{2} + \left\{ {x_{k\; 2} - {{Im}\left\lbrack {\left( {y_{1} + {y_{2}j}} \right)^{{- j} \times {PN}_{}}} \right\rbrack}} \right\}^{2}}{2\delta^{2}}}}}$where NS denotes a number of phase noise samples and a correspondingphase noise sample is denoted as PN_(i). x_(k1) and x_(k2) denote firstand second coordinates of point x_(k), y₁ and y₂ denote coordinates ofthe point y.
 9. The method of claim 1, comprising find the likelihoodfunction (LL) of partition P(Â_(m))={S_(i); i=1, . . . , N}, as${LL}_{m} = {{{LL}\left( \left\{ {{\hat{A}}_{m},{P\left( {\hat{A}}_{m} \right)}} \right\} \right)} = {n^{- 1}{\sum\limits_{k = 0}^{n - 1}\; {\max\limits_{y \in {\hat{A}}_{m}}\; {{LL}\left( {x_{k},y} \right)}}}}}$10. A method to design signal constellation for transmission overcoherent optical channels with nonlinear phase noise, comprising:selecting a signal constellation to be used for initialization, whereinM is a size of the constellation; generating a training sequence from anoptimum source distribution as {x_(j); j=0, . . . , n−1}; groupingsamples from the training sequence into M clusters; determiningmembership to the cluster based on LLR of sample point and candidatesignal constellation points from a previous iteration; assigning eachsample point to the cluster with the largest LLR; determining new signalconstellation points as center of mass for each cluster; and repeatinguntil convergence.
 11. The method of claim 10, comprising given the m-thsubset (cluster) with N candidate constellation points, denoted asÂ_(m)={y_(i); i=1, . . . , N}, finding a likelihood function ofpartition P(Â_(m))={S_(i); i=1, . . . , N}, as${LL}_{m} = {{{LL}\left( \left\{ {{\hat{A}}_{m},{P\left( {\hat{A}}_{m} \right)}} \right\} \right)} = {n^{- 1}{\sum\limits_{k = 0}^{n - 1}\; {\max\limits_{y \in {\hat{A}}_{m}}\; {{LL}\left( {x_{k},y} \right)}}}}}$12. The method of claim 10, comprising generating a function LL(x_(j),y) is the cumulative log-likelihood function defined as $\begin{matrix}{I,I,{\left( {x_{k},y} \right) = {\frac{1}{NS}{\sum\limits_{i = 1}^{NS}\; {- \frac{\begin{matrix}{\left\{ {x_{k\; 1} - {{Re}\left\lbrack {\left( {y_{1} + {y_{2}j}} \right)^{{- j} \times {PN}_{}}} \right\rbrack}} \right\}^{2} +} \\\left\{ {x_{k\; 2} - {{Im}\left\lbrack {\left( {y_{1} + {y_{2}j}} \right)^{{- j} \times {PN}_{}}} \right\rbrack}} \right\}^{2}\end{matrix}}{2\delta^{2}}}}}}} & \;\end{matrix}$ where NS denotes a number of phase noise samples and acorresponding phase noise sample is denoted as PN_(i). x_(k1) and x_(k2)denote first and second coordinates of point x_(k), y₁ and y₂ denotecoordinates of the point y.
 13. The method of claim 1, comprisingfinding a maximum of cumulative log-likelihood function as anoptimization criterion.
 14. The method of claim 1, comprising thechannels with linear and nonlinear phase noise.
 15. The method of claim1, comprising employing LDPC-coded-modulation using signalconstellations from LLR-OSCD.
 16. The method of claim 1, comprisingcommunicating over channel dominated by laser phase noise, imperfectCPE, and nonlinear phase noise introduced by fiber nonlinearities. 17.The method of claim 1, comprising encoding b independent data by an(n,k) LDPC encoder and writing in row-wise fashion into a b×n blockinterleaver.
 18. The method of claim 1, comprising performing MonteCarlo integration at a receiver.
 19. The method of claim 1, comprisinggenerating an optimum source by maximizing channel capacity based onArimoto-Blahut method.